Problem: Archimedes went to sleep beside a big rock. He wanted to get up at $7$ AM, but the alarm clock was yet to be invented! He decided to sleep at the spot where the rock's shadow should end when it's $7$ AM so as to be awakened by the direct sunlight. Archimedes knew that at $7$ AM, the sunlight reaches the ground at an angle of $31^\circ$. The rock beside which he slept was $5$ meters tall. How far from the rock did Archimedes go to sleep? Round your final answer to the nearest hundredth.
Answer: The strategy Model the situation as a right triangle. Determine the appropriate trigonometric ratio in order to find the missing side. Form an equation and solve for the missing side. Calculate the final result and round. Modeling as a right triangle This situation can be modeled by the following right triangle. The height is $5\text{ m}$ and the angle on the right is $31^\circ$. We are asked to find the distance between Archimedes and the rock, which is the base of the triangle. ${31^{\circ}}$ $5$ $?$ Determining the appropriate trigonometric ratio We are given the measure of an angle and the length of the side ${\text{opposite}}$ to the given angle. We are asked to find the side ${\text{adjacent}}$ to the given angle. The appropriate trigonometric ratio is therefore the $\text{tangent}$. Forming an equation and solving Denoting the missing side by $x$, we obtain the equation $\tan(31^\circ)=\dfrac{5}{x}$. Solving the equation, we get $x=\dfrac{5}{\tan(31^\circ)}$. Evaluating this result in the calculator and rounding to the nearest hundredth, we get $x=8.32\text{ m}$. Summary Archimedes went to sleep $8.32$ meters from the rock.